See the attachment.
If you have trouble visualising the curve you can see the animation
motion on a saddle on my web site Calculus 3
Let i + j + k, and be the curve of intersection of the hyperbolic paraboloid and the cylinder oriented counterclockwise as viewed from above.
(i) Obtain a parametrization of the curve and use the result to evaluate r.
(ii) use Stokes's Theorem to evaluate r.